4. Momentarily not worrying about whether n is even or odd, let n = 2t, we have

[(at)2 + (bt)2)]1/(2t) < [ (at)2 + (bt)2)]1/2

5. Or,

[an + bn]1/n <
[an + bn]1/2

6. And statement 3. is established.

7. A standard proof of the law of cosines places the longest side c, off the x axis.

8. We shall use this law, but also establish another relation by placing c on the x axis and the altitude of the triangle ABC (the capitalized letters represent the vertices) on the y axis. We call the origin 0, so that C0 is the altitude, which we call h.
We say the length BO = r and the length 0A = c - r. Arbitrarily, a < b < c, with a =/= 0.

8a. That is, we have two right triangles joined by h, which is perpendicular to c. Under side b on the x axis is base r and under side a on the x axis is (c - r).